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Chapter 1: Problem 14

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$3 x+11<5$$

Short Answer

Expert verified
The solution to the inequality is \((-\infty, -2)\).

Step by step solution

01

Isolate the variable

To solve the inequality \(3x + 11 < 5\), we need to isolate \(x\) on one side. Start by subtracting 11 from both sides of the inequality.\[3x + 11 - 11 < 5 - 11\]This simplifies to:\[3x < -6\]
02

Solve for x

To isolate \(x\), divide both sides by 3.\[\frac{3x}{3} < \frac{-6}{3}\]Simplifying gives:\[x < -2\]
03

Express the solution in interval notation

The solution \(x < -2\) can be written in interval notation as:\((-\infty, -2)\) which indicates all numbers less than -2.
04

Graph the solution set

On a number line, shade all numbers to the left of -2 and use an open circle at -2 to show that -2 is not included in the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate the Variable
When faced with a linear inequality such as \(3x + 11 < 5\), your first task is to isolate the variable \(x\). This means you need to manipulate the inequality until \(x\) stands alone on one side. To begin:
  • Subtract 11 from both sides of the inequality.
  • That transforms the inequality to \(3x < -6\).
The goal is to have \(x\) by itself, which will make it simpler to see the values \(x\) can hold. Next, divide each side by 3 to finish isolating \(x\). This step ensures that you've simplified properly, resulting in \(x < -2\).
Remember, if ever you multiply or divide by a negative number, you have to flip the inequality sign. However, in this problem, since we divided by a positive number, the inequality sign remains the same.
Interval Notation
Once you've established that \(x < -2\), the next step is to express this solution using interval notation. This notation allows you to write the solution in a compact form. It uses parentheses and brackets to indicate a range of values.
  • Since \(x\) is less than, but not equal to -2, we denote this with an open parenthesis \(()\).
  • The notation \((-\infty, -2)\) tells us that \(x\) includes all numbers less than -2.
Whenever you use \(-\infty\) or \(\infty\), always use parentheses because you can never "reach" infinity, hence it can't be closed with a bracket. Interval notation is crucial for simplifying the expression of our solutions.
Graphing Inequalities
Graphing inequalities visually demonstrates the range of possible solutions. For \(x < -2\), you can easily represent it on a number line:
  • Locate -2 on the number line.
  • Draw an open circle at -2 to indicate it's not included in the solution.
  • Shade the entire region to the left of -2, which signifies all the numbers less than -2 fit our inequality.
The open circle is key in distinguishing inequalities like \(<\) or \(>\), thereby contrasting from inequalities such as \(\leq\) or \(\geq\) where you would use a closed circle. Graphing is a handy tool for visual learners, helping them understand which values make the inequality true.

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Most popular questions from this chapter

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